Starting from the natural symplectic structures determined by the Weyl group for the single massive spinning particle, we build the composed system of two spinning particles. The groupal analysis allows us to determine the centre-of-mass co-ordinates, to which we add a set of complementary ones in such a way as to get collective and relative canonical variables of momentum and position. Exploiting these results we are able to introduce an analogous description of the phase-space of the scalar three-body. The equations of motion are obtained by means of the full foliation on the state equations and we are able to get them in a purely Hamiltonian scheme on the reduced physical phase-space. For the three-body we get three first-class constraints with two independent potentials; however, the Hamiltonian scheme suggests a more flexible method for introducing interactions. We remark that the direct canonical quantization, allowed by the Hamiltonian framework, gives rise, in the two-scalar-particle case with the Coulomb potential, to a Lamb shift term and with the rising linear potential to very good results about the mesons properties. Indeed, we find a wave equation formally coinciding with the one obtained and studied in the domain of the anisotropic chromodynamics.